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Properties

Label 12.0.82587655432...4677.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 13^{11}\cdot 43^{6}$
Root discriminant $119.24$
Ramified primes $3, 13, 43$
Class number $114320$ (GRH)
Class group $[2, 2, 2, 14290]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17208116641, -3249472929, 3249472929, -196019617, 196019617, -5178785, 5178785, -66977, 66977, -417, 417, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 417*x^10 - 417*x^9 + 66977*x^8 - 66977*x^7 + 5178785*x^6 - 5178785*x^5 + 196019617*x^4 - 196019617*x^3 + 3249472929*x^2 - 3249472929*x + 17208116641)
 
gp: K = bnfinit(x^12 - x^11 + 417*x^10 - 417*x^9 + 66977*x^8 - 66977*x^7 + 5178785*x^6 - 5178785*x^5 + 196019617*x^4 - 196019617*x^3 + 3249472929*x^2 - 3249472929*x + 17208116641, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 417 x^{10} - 417 x^{9} + 66977 x^{8} - 66977 x^{7} + 5178785 x^{6} - 5178785 x^{5} + 196019617 x^{4} - 196019617 x^{3} + 3249472929 x^{2} - 3249472929 x + 17208116641 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8258765543212396597594677=3^{6}\cdot 13^{11}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.24$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 13, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1677=3\cdot 13\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1677}(128,·)$, $\chi_{1677}(1,·)$, $\chi_{1677}(259,·)$, $\chi_{1677}(644,·)$, $\chi_{1677}(517,·)$, $\chi_{1677}(646,·)$, $\chi_{1677}(1289,·)$, $\chi_{1677}(1291,·)$, $\chi_{1677}(1420,·)$, $\chi_{1677}(515,·)$, $\chi_{1677}(902,·)$, $\chi_{1677}(773,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1854584161} a^{7} - \frac{802436458}{1854584161} a^{6} + \frac{224}{1854584161} a^{5} - \frac{137314573}{1854584161} a^{4} + \frac{14336}{1854584161} a^{3} + \frac{827237140}{1854584161} a^{2} + \frac{229376}{1854584161} a + \frac{112757828}{1854584161}$, $\frac{1}{1854584161} a^{8} + \frac{256}{1854584161} a^{6} - \frac{286211598}{1854584161} a^{5} + \frac{20480}{1854584161} a^{4} + \frac{570748345}{1854584161} a^{3} + \frac{524288}{1854584161} a^{2} - \frac{281894570}{1854584161} a + \frac{2097152}{1854584161}$, $\frac{1}{1854584161} a^{9} - \frac{721320221}{1854584161} a^{6} - \frac{36864}{1854584161} a^{5} + \frac{486179974}{1854584161} a^{4} - \frac{3145728}{1854584161} a^{3} - \frac{632008056}{1854584161} a^{2} - \frac{56623104}{1854584161} a + \frac{807342608}{1854584161}$, $\frac{1}{1854584161} a^{10} - \frac{46080}{1854584161} a^{6} + \frac{713087471}{1854584161} a^{5} - \frac{4915200}{1854584161} a^{4} + \frac{907982625}{1854584161} a^{3} - \frac{141557760}{1854584161} a^{2} - \frac{516984750}{1854584161} a - \frac{603979776}{1854584161}$, $\frac{1}{1854584161} a^{11} - \frac{714479312}{1854584161} a^{6} + \frac{5406720}{1854584161} a^{5} - \frac{560968044}{1854584161} a^{4} + \frac{519045120}{1854584161} a^{3} - \frac{552418744}{1854584161} a^{2} + \frac{692745499}{1854584161} a - \frac{664104882}{1854584161}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{14290}$, which has order $114320$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 120.784031363 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.36560277.4, \(\Q(\zeta_{13})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$43$43.6.3.1$x^{6} - 86 x^{4} + 1849 x^{2} - 7950700$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43.6.3.1$x^{6} - 86 x^{4} + 1849 x^{2} - 7950700$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$